Axiom ax-mulass 8041

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7999. Proofs should normally use mulass 8069 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7936	. . . 4 class ℂ
3	1, 2	wcel 2177	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2177	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2177	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 981	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7943	. . . . 5 class ·
10	1, 4, 9	co 5954	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5954	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5954	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5954	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1373	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8069